pubs.acs.org/Langmuir © 2010 American Chemical Society
Direct Measurement of Interface Anisotropy of Bicontinuous Structures via 3D Image Analysis Carlos R. Lopez-Barron*,† and Christopher W. Macosko‡ †
Chemical Engineering, University of Delaware, Newark, Delaware 19716, and ‡Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455 Received June 7, 2010. Revised Manuscript Received July 21, 2010
A very important morphological parameter in two-phase fluids is the interface anisotropy, which can be quantified using the interface tensor, qij. However, the computation of this tensor for complex interfaces is not straightforward. A novel method (the local cross product method, LCPM) to compute the interface tensor of two-phase fluids using 3D imaging coupled with differential geometry is presented here. The method was used to evaluate the degree of anisotropy of phase separated systems with bicontinuous morphologies subjected to uniaxial and shear deformation fields. A model bicontinuous structure (i.e., the gyroid surface) was used to assess the accuracy and precision of the method. The method was then used to track the anisotropy changes of an immiscible polymer blend with cocontinuous morphology, during uniaxial deformation and subsequent retraction. It was found that the dependence of the anisotropy on the Hencky strain of both the gyroid surface and the cocontinuous blend follow the same trend. The retraction of the blend after uniaxial extension is accompanied by an exponential decay of the second invariant of qij, which obeys the relation: |IIq|/Q2 ∼ e-0.129t.
Introduction Microstructure plays an important role in determining the properties of multiphase polymeric systems, such as polymer blends,1-4 and self-assembled block copolymers.5,6 Hence, a complete and accurate characterization of the morphology (i.e., measurement of size, shape and orientation of the phases or the interface) is crucial for the design and study of these materials. Typically, a comprehensive morphological characterization involves a combination of direct and indirect methods. Traditional direct methods (including electronic, optical and atomic force microscopy3,7) are limited to 2D cut projections of the complete 3D microstructure, with inherent measurement errors due to user bias (e.g., mistaking transversally cut rods for droplets) and limited spatial averaging. On the other hand, indirect methods (small-angle neutron, X-ray and light scattering techniques8,9 and rheological measurements10) give good global averaging statistical measurement but their interpretation often relies on good models, usually based on the direct methods. 3D microscopy offers the benefits of both direct and indirect methods, i.e., direct visualization with good global statistical average without obscuring any morphological feature and without the need of any model for interpretation. *Corresponding author. E-mail:
[email protected]. (1) Cook, W. D.; Zhang, T.; Moad, G.; Deipen, G. V.; Cser, F.; Fox, B.; O’Shea, M. J. Appl. Polym. Sci. 1996, 62, 1699–1708. (2) Knackstedt, M. A.; Roberts, A. P. Macromolecules 1996, 29, 1369–71. (3) Harrats, C.; Thomas, S.; Groeninckx, G. Micro and Nanostructured Polymer Blends: Phase Morphology and Interfaces; Taylor Francis: Boca Raton, FL, 2005. (4) Omonov, T. S.; Harrats, C.; Groeninckx, G.; Moldenaers, P. Polymer 2007, 48, 5289–5302. (5) Dair, B. J.; Avgeropoulos, A.; Hadjichristidis, N.; Thomas, E. L. J. Mater. Sci. 2000, 35, 5207–5213. (6) Meuler, A. J.; Fleury, G.; Hillmyer, M. A.; Bates, F. S. Macromolecules 2008, 41, 5809–5817. (7) Sawyer, L. C.; Grubb, D. T.; Meyers, G. F. Polymer Microscopy; Springer: New York, 2008. (8) Roe, R. J. Methods of X-ray and Neutron Scattering in Polymer Science; Oxford University Press: New York, 2000. (9) Hashimoto, T.; Takenaka, M.; Jinnai, H. J. Appl. Crystallogr. 1991, 24, 457– 466. (10) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: New York, 1999.
14284 DOI: 10.1021/la102314r
3D imaging coupled with differential geometry has been recently used to quantify the size (via the measurement of interfacial area), shape (via the computation of local interfacial curvature) and topology of polymer systems11-13 and metal alloys14,15 with bicontinuous interfaces. However, the interface anisotropy (global orientation) of these types of systems has never been computed before. In this work, we present the first method to compute interface anisotropy from 3D images of two phase systems with bicontinuous morphologies. However, the method is not restricted to any particular type of morphology. Being able to compute interface anisotropy is very useful in the computation of interfacial stresses, and more specifically, in understanding the rheological behavior of biphasic systems. Doi and Ohta16 showed that, in mixtures of immiscible fluids, the contribution to the stress tensor due to the interface is given by the product Γqij, where Γ is the interfacial tension, and qij is the interface tensor, defined as Z 1 1 ð1Þ ni nj - δij dS qij ¼ V S 3 where ni is the ith component of the unit normal vector to the interface, δij is the Kronecker’s delta and the integral is performed on the surface S contained in the volume V. Doi and Ohta described the microstructure of the mixtures with the tensor qij and the interfacial area per unit volume, Q, given as Z 1 Q ¼ dS ð2Þ V S They also defined the degree of anisotropy as qij/Q. They proposed phenomenological kinetic equation for the time evolution (11) Nishikawa, Y.; Jinnai, H.; Koga, T.; Hashimoto, T.; Hyde, S. T. Langmuir 1998, 14, 1242–1249. (12) Nishikawa, Y.; Koga, T.; Hashimoto, T.; Jinnai, H. Langmuir 2001, 17, 3254–3265. (13) Lopez-Barron, C.; Macosko, C. W. Langmuir 2009, 25, 9392–9404. (14) Kwon, Y.; Thornton, K.; Voorhees, P. W. Phys. Rev. E 2007, 75, 021120. (15) Mendoza, R.; Thornton, K.; Savin, I.; Voorhees, P. W. Acta Mater. 2006, 54, 743–750. (16) Doi, M.; Ohta, T. J. Chem. Phys. 1991, 95, 1242–1248.
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of qij and Q, which they used to derive a rheological constitutive equation of concentrated biphasic fluids with complex interfaces.16 Wetzel and Tucker17 proposed a microstructural variable that includes both qij and Q, namely, the area tensor Z 1 ni nj dS ð3Þ Aij ¼ V S 1 ¼ qij þ Qδij 3
ð4Þ
The traces of Aij and qij are Q and 0, respectively. Both tensors are real and symmetric. Consequently, their eigenvalues are real and their eigenvectors are orthogonal. The eigenvectors of either Aij or qij, indicate the most and least likely orientations of the interfacial area, and their eigenvalues give the relative amounts of interfacial area oriented in each principal direction.17 Despite the importance of the tensor qij, it has been calculated only for few simple instances.17-20 Evidently, obtaining qij (or Aij) requires the computation of the interface normal vector ni. The three components of ni are not available from 2D micrographs without previous knowledge (or assumption) of the interface shape. Considering cylindrical deformed drops with hemispherical end-caps, Vinckier et al. computed the components of qij for droplets deformed into fibrils.18 Similarly, Wetzel and Tucker gave the components for ellipsoidal droplets.17 Almusalam et al. computed the components for elongated droplets (threads) during the growth of disturbances and thread break up.19 Using the axisymmetry of the elongated droplets Almusallam et al. developed exact expressions for the components of qij. Takahashi and Okamoto developed analytical expressions for the excess shear stress, Δσxy = -Γqxy, for different droplet shapes, namely, ellipsoid, rod-like shape, dumbbell, and ellipsoid of revolution.20 Interfaces in cocontinuous morphologies, formed during spinodal decomposition9,21,22 or mechanical mixing of two immiscible liquids,23-25 are considerable more complex than those in matrixdroplet morphologies. They are not axisymmetry and they cannot be described by any analytical equation. Therefore, the approach followed in previous works to compute qij is not applicable for this kind of morphology. In this paper we present a novel method, hereafter referred as the local cross product method (LCPM), to compute the interface tensor and the surface tensor from 3D images. This method was tested with a model bicontinuous surface, namely, the gyroid surface. Good agreement was found between the anisotropy computed with LCPM and that computed analytically as well as with the predictions from Doi-Ohta’s theory.16 Additionally, LCPM was applied to measure the anisotropy evolution of a model blend made of fluorescently labeled polystyrene (FLPS) and styrene-acrylonitrile copolymer (SAN) during uniaxial elongation and subsequent relaxation.
The Local Cross Product Method (LCPM) Measurement of the Normal Vector Field. Given a triangular polygon with three vertices, pn (xn, yn, zn) (with n = 1,2,3), the (17) Wetzel, E. D.; Tucker, C. L. Int. J. Multiphase Flow 1999, 25, 35–61. (18) Vinckier, I.; Mewis, J.; Moldenaers, P. Rheol. Acta 1997, 36, 513–523. (19) Almusallam, A. S.; Larson, R. G.; Solomon, M. J. J. Non-Newtonian Fluid Mech. 2003, 113, 29–48. (20) Takahashi, M.; Okamoto, K. J. Soc. Rheol., Jpn. 2007, 35, 199–205. (21) Jinnai, H.; Koga, T.; Nishikawa, Y.; Hashimoto, T.; Hyde, S. T. Phys. Rev. Lett. 1997, 78, 2248–2251. (22) Jinnai, H.; Nishikawa, Y.; Ikehara, T.; Nishi, T. Adv. Polym. Sci. 2004, 170, 115–167. (23) Potschke, P.; Paul, D. R. J. Macromol. Sci., Polym. Rev. 2003, C43, 87–141. (24) Pyun, A.; Bell, J. R.; Won, K. H.; Weon, B. M.; Seol, S. K.; Je, J. H.; Macosko, C. W. Macromolecules 2007, 40, 2029–2035. (25) Bell, J. R.; Chang, K.; Lopez-Barron, C. R.; Macosko, C. W.; Morse, D. C. Macromolecules 2010, 43, 5024–5032.
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Figure 1. (a) Schematic of the computation of the vector normal to a triangle. (b) Detail of the normal vector field generated with computation scheme (part a) on a cocontinuous interface.
normal to that triangle can be obtained by first describing two directional vectors in the same plane, e.g., d1 = (x2 - x1,y2 - y1, z2 - z1) and d2 = (x3 - x2,y3 - y2,z3 - z2) (see Figure 1a). The cross product d1 d2 gives a vector perpendicular to the polygon. Therefore, the unit normal vector is given by ^n ¼
d1 d2 jd1 d2 j
ð5Þ
Accordingly, given a surface that we can be represented with a triangular mesh, the normal vector field is readily obtainable by performing the cross product (eq 5) on each triangle in the mesh. Triangular Mesh Generation. 3D images are commonly obtained from the reconstruction of sets of tomograms (e.g., from X-ray tomography24,26 or TEM tomography27) or optically sectioned slices (e.g., from laser scanning confocal microscopy (LSCM)11-13). In this work, LSCM was used to image the FLPS/SAN cocontinuous blend. The contrast in the LSCM micrographs was achieved by using a detector barrier filter (461 nm) to detect only fluorescence (26) Momose, A.; Fujii, A.; Kadowaki, H.; Jinnai, H. Macromolecules 2005, 38, 7197–7200. (27) Jinnai, H.; Spontak, R. J.; Nishi, T. Macromolecules 2010, 43, 1675–1688.
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Figure 2. Gyroid surface computed with eqs 9 or 10, with different degrees of uniaxial extension or shear deformation. The plotted range for all the images is -π e x, y, z e π.
from the fluorescent label (anthracene) attached to the polystyrene. On the other hand, 2D slices of the gyroid surface were obtained analytically, as described below. The 2D images of the cocontinuous structures were deconvoluted and thesholded to obtain black and white images, as described in a recent paper.13 The 3D reconstruction was performed by applying a nonstructuring meshing method based on the marching cubes algorithm (MCA).28 This algorithm produces a triangle mesh by computing iso-surfaces from a three-dimensional scalar field (voxels), i.e., from the 2D (thresholded) slices data. The number of triangles per image is of the order of 107. The unit normal vector of each triangle was calculated with eq 5. Figure 1b shows a detail of the normal vector field for a cocontinuous interface. Numerical Procedure. Once obtained the normal vector field, the surface tensor is computed with the discrete version of eq 1, that is N 1X 1 ni, k nj, k - δij Ak ð6Þ qij ¼ V k¼1 3 where N is the total number of triangles. ni,k and Ak are the ith component of the normal vector and the area of the kth triangle, respectively. Similarly, the specific interfacial area was computed with
minimal surface with crystallographic order. Exact expressions for the gyroid surface can be obtained using the EnneperWeierstrass representation.30,31 However, these expressions are very complex mathematically and very hard to handle analytically. Alternately, the gyroid surface, with crystallographic unit cell length L=2π, can be generated using the trigonometric approximation obtained by Barnes32 Fðx, y, zÞ ¼ sin x cos y þ sin y cos z þ sin z cos x ¼ 0
ð8Þ
The gyroid surface is globally isotropic. We induced anisotropy by analytically applying shear or uniaxial deformation to the surface. Considering a cubic portion of a gyroid, described by eq 8, with volume L3, the expression for the sheared (in x-direction) surface is given as Fðx þ γ, y, zÞ ¼ 0
ð9Þ
and for uniaxially elongated surface (in z-direction) Fðλ - 1=2 x, λ - 1=2 y, λzÞ ¼ 0
ð10Þ
The gyroid surface (Figure 2) was used to test the accuracy of LCPM. This surface (discovered by Schoen29) is a triply periodic
where γ is the shear strain and λ is the uniaxial stretch. The prefactor λ-1/2 arises from the constant volume condition, namely, x 3 y 3 z=L3 =λ-1/2x 3 λ-1/2y 3 λz. In this paper we used the Hencky strain, which is defined as: ε = ln λ. An important question to be asked is: what is the effect of the interface deformation on the amount of interface (i.e., on Q)? Figure 3 shows that a monotonic increase of Q is observed for both uniaxial and shear deformation, and becomes important at ε or γ > 1. This is an important finding because the interfacial stress depends not only on the interface anisotropy but also on
(28) Lorensen, W. E.; Cline, H. E. SIGGRAPH Comput.Graph. 1987, 21, 163– 169. (29) Schoen, A. H. In Infinite Periodic Minimal Surfaces Without Self-Intersections; NASA Tech. Note No. D-5541; NASA: Washington, DC, 1970.
(30) Nitsche, J. C. C. Lectures on Minimal Surfaces; Cambridge University Press: Cambridge, U.K., and New York, 1988. (31) Gandy, P. J. F.; Klinowski, J. Chem. Phys. Lett. 2000, 321, 363–371. (32) Barnes, I. S. Austr. Math. Soc. Gaz. 1990, 17, 99–105.
Q ¼
N 1X Ak V k¼1
ð7Þ
Assessment of the LCPM with a Model Bicontinuous Structure
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Figure 3. Specific interfacial area as a function of shear strain or Hencky strain for the gyroid surface. Symbols give results computed with eq 7 and solid lines give theoretical predictions (eqs 17 and 21 in Appendix A).
the value of Q.16,33,34 It is noteworthy that the interfacial area computed from the triangular mesh using eq 7 is in perfect accordance with the theoretical predictions using the second invariant of the finger tensor35 which gives the relations Q ¼ Q0 ½1=3 ð2λ þ λ - 2 Þ1=2 and Q ¼ Q0 ½1 þ 1=3 γ2 1=2 for uniaxial extension and shear deformation, respectively (see Appendix A for details). It is worthwhile to note that the good agreement between the interfacial area computed via eq 7 and eqs 17 and 21 is an indication that the gyroid surfaces analyzed here are of good quality. Nishikawa et al.11,12 measured the quality of 3D interfaces via the roughness index, defined as RI ¼ ðÆAtri æÞ1=2 ðÆK1 æ þ ÆK2 æÞ=2 where ÆAtriæ, Æκ1æ, and Æκ2æ denote the average area and principal curvatures of the triangular mesh. We recently demonstrated that the interface curvatures of the gyroid surface with RI = 0.08 give errors within 6% when they are computed via the coordinate transformation method.13 The surfaces analyzed in this paper have RI = 0.08. This surface roughness allowed the very accurate computation of specific interfacial area (Figure 3) and of the components of the interface tensor for strains >0.01 (see Appendix B). The normal vector field of the deformed surfaces was computed using two methods: 1 The first method consisted on first generating 2D cuts of the gyroid (as if they were visualized with the confocal microscope). This was achieved by fixing the value of the z-coordinate (z = h, - π e h e π) and plotting eqs 9 or 10 bidimensionally (in the plane x-y). Once obtained the stack of 2D images, triangular meshes of the gyroid surface were generated using MCA and both the normal vector field and the surface tensor were computed with LCPM, i.e. with eqs 5 and 6, respectively. 2 In the second method, the surface tensor was analytically computed. From basic geometry we know that for a surface given implicitly as a set of points (x, y, z), satisfying F(x,y,z) = 0, a normal on the surface at a point (x, y, z) is given by the gradient of F. Hence the unit normal vector field is simply rF n^ ¼ jrFj
ð11Þ
(33) Lee, H. M.; Park, O. O. J. Rheol. 1994, 38, 1405–1425. (34) Tucker, C. L.; Moldenaers, P. Annu. Rev. Fluid Mech 2002, 34, 177–210. (35) Macosko, C. W. Rheology: Principles, Measurements, and Applications; VCH: New York, 1993.
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Figure 4. (a) Diagonal and (b) off-diagonal components of the interface tensor as a function of the Hencky strain for the gyroid surface subjected to uniaxial extension. Solid lines and symbols give results computed analytically and with the LCPM, respectively.
where F is given by either eqs 9 or 10. Substituting eq 11 into eq 1 gives a very complex expression, difficult to integrate analytically. Thus, the interface tensor was computed by considering uniform (1 pixel2) subareas of the interface and integrating numerically eq 1 using the values of ni computed with eq 11 in every subarea where the interface is defined (i.e., where eqs 9 or 10 are satisfied). Gyroid Surface under Uniaxial Extension. Examples of the gyroid surface subjected to uniaxial extension are shown in Figure 2. The components of qij computed with the two methods described above are shown in Figure 4. The dependence of the components with the applied strain computed analytically is well predicted using the LCPM. However, at high values of ε, the LCPM overestimates the values of the off-diagonal components. This could be due to the nonuniformity of the triangle sizes generated with MCA when the interface is extremely elongated (i.e., when ε >1). It is noteworthy that the quantity (qzz - qxx)/Q0 predicted by Doi and Ohta’s theory16 is in very good agreement with the corresponding values computed with the LCPM (Appendix B). The of the interface tensor of being traceless17 Pcharacteristic 3 (i.e., i=1qij = 0) is fulfilled, as evidenced in Figure 4a. The exponential growth of qxy with ε attests the increase of interface anisotropy. In order to elucidate the origin of such anisotropy, which is not evident in Figure 4b, we performed an statistical analysis of the components of the normal vector field, ^n, and of the integrands in eq 1 for the off-diagonal components of qij. We defined the probability density of the components of the normal vector field as: N P
Pni ðni Þ ¼
k¼1
A½kjni - Δni =2 e ni, k < ni þ Δni =2 Δni
N P k¼1
ð12Þ Ak
where Ak and ni,k are the area and the ith normal vector component of the kth triangle, N is the total number of triangles, and Δni is the bin size. The term A[k|ni - Δni/2 e ni,k < ni þ Δni/2] represents the surface area of the triangles satisfying the condition: DOI: 10.1021/la102314r
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ni - Δni/2 e ni,k < ni þ Δni/2. Similarly, the probability density of the products of the components of ^n, Pninj(ninj), are computed simply by replacing ni for ninj in eq 12 The probability densities of the three components of the normal vector field and their products are symmetric and zerocentered (see Figure 5). These characteristics indicate that the interface is globally equivalent to a close surface, that is, eachnormal vector on the interface corresponds to a vector pointing in the opposite direction. The fact that all three distributions are identical is a clear indication of isotropy. That is to say, the normal vector field points evenly in all directions and the three offdiagonal components of qij are equal. When uniaxial extension is applied, the amount of interface pointing in the direction of deformation (z-direction in this case) decreases in favor of an increase of the interface pointing laterally. This is evident in the elongated surfaces shown in Figure 2 and in the probability densities of the components of ^n, shown in Figure 6, parts a and b. This produces different distributions of the products ninj with bigger populations of nxny (see Figure 6, parts c and d), which explain the growth of qxy and decrease of qxz and qyz shown in Figure 4b. Doi and Ohta defined the degree of anisotropy as the quantity qij/Q. However, being this quantity a tensor, its interpretation is not straightforward without the analysis of its components and their interrelations. In order to come up with a single quantity that describes the global anisotropy we considered the invariants of the interface tensor, which are defined as Iqij = tr(qij), IIqij = 1/2(Iqij2 tr(qij2)), and IIIqij = det(qij). Figure 7 shows the three invariants normalized with the first, second and third power of Q, respectively, as a function of the Hencky strains for the uniaxially extended gyroid surface. As expected, due to the traceless characteristic of qij, the first invariant is zero regardless of the deformation applied. Both the second and third invariants increase with the applied strain. Nevertheless, IIqij is more sensitive to changes in anisotropy than IIIqij. Hence, we consider IIqij a measure for anisotropy. The reason why the second invariant is more sensitive escapes from our understanding and is beyond the scope of this paper, but deserves further analysis. At low deformations (ε < 0.1) the anisotropy is negligible, but at ε > 0.1, it grows rapidly until it reaches a plateau. At this stage (i.e., after the onset of the plateau) practically all the interface is directed perpendicular to the z-axis; therefore, further deformation does not produces more anisotropy. The above pattern is well reproduced by the LCPM, however when compared to the analytical results, errors of up to 13% are observed in the high deformation regime. In the low deformation zone the error was within 3% (see inset in Figure 7). Gyroid Surface under Shear Deformation. The components of qij computed analytically and with the LCPM for the sheared gyroid surface are shown in Figure 8. Similar to the uniaxial extension case, the dependence of the qij-components with strain computed with LCPM is qualitatively in agreement with the analytical values, but their numerical values are slightly overestimated in the regime of high strains. Surprisingly the dimensionless quantities (qxx - qzz)/Q0 and qxz/Q0 computed with the LCPM are in very good agreement with those predicted by the Doi and Ohta’s theory (see Appendix B). Unlike the uniaxial extension case, each of the off-diagonal components evolves differently. As shown in Figure 2, by shearing the surface in the y-direction (with the velocity gradient in the z-direction and vorticity in the x-direction) the surface is aligned in such a way that the population of components of the normal vector in the z-direction grows while that in the y-direction decreases and that in the x-direction remains unchanged. This is clear in the proba14288 DOI: 10.1021/la102314r
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Figure 5. Probability densities of (a) the components of the normal vector and (b) their products contributing to the off-diagonal components of qij. Data are given for the gyroid surface under no deformation.
bility densities shown in Figure 9 (a and b). These changes produces different probability distributions of the products ninj (i,j= x,y,z) with a notable asymmetric distribution of nxnz (Figure 9, parts c and d). This asymmetry augments with the applied strain, which results in the growth of qxy with γ. The three invariants of qij are plotted as a function of γ in Figure 10. Same as in the uniaxial extension case, the second invariant is the most sensitive to anisotropy changes. The constant zero value of the first invariant is, again, due to the characteristic of qij of being traceless. LCPM reproduces the evolution of the invariants with good qualitative agreement with the analytical result. However, a numerical error of up to 18% is observed for high shear strains (γ > 0.2). The error is within 5% for γ < 0.2 (see inset in Figure 10). It is well-known that one of the self-assembly morphology observed in block copolymer is the gyroid.36 Accordingly, being aware of the errors in the different strain zones, the results of anisotropy change during elongation and shear of the model gyroid surface presented above are potentially very useful to model the rheological behavior of systems with bicontinuous (or cocontinuous) morphologies such as block copolymers37,38 and immiscible blends.16,33 Likewise, direct comparison of anisotopy obtained from indirect methods, such as SAXS39,40 or SALS,37 with real 3D-space measurements will be possible with the method (LCPM) proposed here. Additionally this method could be used to extend the analysis of stress-anisotropy during droplet retraction, (36) Hajduk, D. A.; Harper, P. E.; Gruner, S. M.; Honeker, C. C.; Kim, G.; Thomas, E. L.; Fetters, L. J. Macromolecules 1994, 27, 4063–4075. (37) Krishnan, K.; Chapman, B.; Bates, F. S.; Lodge, T. P.; Almdal, K.; Burghardt, W. R. J. Rheol. 2002, 46, 529–554. (38) Zhou, N.; Bates, F. S.; Lodge, T. P.; Burghardt, W. R. J. Rheol. 2007, 51, 1027–1046. (39) Caputo, F. E.; Burghardt, W. R.; Krishnan, K.; Bates, F. S.; Lodge, T. P. Phys. Rev. E 2002, 66, 041401. (40) Zhou, N.; Lodge, T. P.; Bates, F. S. Soft Matter 2010, 6, 1281–1290.
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Figure 6. Probability densities of (a and b) the components of the normal vector and (c and d) their products contributing to the off-diagonal components of qij. Data for the gyroid surface were subjected to uniaxial extension with (a and c) ε = 0.2 and (d and d) ε = 1.0.
Figure 7. Normalized invariants of interface tensor as a function of Hencky strain for the gyroid surface. Solid lines and symbols give results computed analytically and with the LCPM, respectively. The inset gives the error generated by the LCPM for the normalized second invariant.
performed by Almusallam and co-workers,19,41 to more complex interfaces.
Application of the LCPM to a Cocontinuous Polymer Blend During Uniaxial Extension and Retraction Experimental Setup. 3D images of the cocontinuous structure of an immiscible blend made of fluorescently labeled polystyrene (FLPS) and styrene-ran-acrylonitrile copolymer (SAN) were obtained with LSCM. The details of the synthesis and characterization of these polymers are described in a previous publication.13 A symmetric (by weight) diblock copolymer made of polystyrene and poly(methyl methacrylate) (PS-b-PMMA) was supplied by Polymer Source and used to stabilize the blend morphology during the extension and retraction experiments.25 (41) Almusallam, A. S.; Larson, R. G.; Solomon, M. J. J. Rheol. 2004, 48, 319– 348.
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Figure 8. (a) Diagonal and (b) off-diagonal components of the interface tensor as a function of the shear strain for the gyroid surface subjected to shear deformation Solid lines and symbols give results computed analytically and with the LCPM, respectively.
Table 1 shows the properties of the blend components for these experiments. A 50/50 FLPS/SAN blend with 1%wt of PS-b-PMMA was prepared by mechanical mixing in a 4 g twin-screw extruder (Microcompounder, DACA Instruments) at 180 C under nitrogen purge. After the mixing, the blends were extruded out of the mixer, quenched with water and molded into rectangular bars (with dimensions: 40 2 1 mm3) . The bars were quiescently annealed at 200 C for 5 min to eliminate microstructure anisotropy generated during the extrusion operation. Subsequently, the bars were subjected to uniaxial extension at 200 C using the TA Instruments Extensional Viscosity Fixture (EVF) attached to the ARES rheometer. A constant extension rate =0.2 s-1 was applied DOI: 10.1021/la102314r
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Figure 9. Probability densities of (a and b) the components of the normal vector and (c and d) their products contributing to the off-diagonal components of qij. Data for the gyroid surface subjected to uniaxial extension with (a and c) γ = 0.2 and (b and d) γ = 1.0. Table 1. Properties of Blend Components polymer FLPS SAN PS-b-PMMA
Figure 10. Normalized invariants of interface tensor as a function of shear strain for the gyroid surface. Solid lines and symbols give results computed analytically and with the LCPM, respectively.
to all the samples. The deformation was stopped at different times ranging from 1 to 10 s, after which, cold water was sprayed on the samples in order to freeze the morphology. A set of elongated samples (those with the maximum deformation) were quiescently annealed at 200 C and submerged in cold water at different times to freeze the morphology. Triangular meshes describing the 3D interface of the deformed/annealed blend microstructure were generated using LSCM and MCA, as described above. Uniaxial Deformation and Relaxation. 3D rendered micrographs of the FLPS/SAN blend with different levels of uniaxial deformation are shown in Figure 11. The alignment of the cocontinuous domains when a Hencky strain of 0.4 is applied (Figure 11b) is very subtle but evident when comparing with the undeformed samples (Figure 11a). As expected, increasing the strain applied the alignment is more dramatic (see Figure 11c). The invariants of the interface tensor (shown in Figure 12) follow the same trends as those observed in the gyroid surface, namely, strain independent zero-valued first invariant, growth of the second and third invariants with IIqij being the most sensitive to anisotropy changes. It is noteworthy that the numerical values 14290 DOI: 10.1021/la102314r
AN-content, % mol 28.4
Mw, kg/mol
Mw/Mn
122 103 50.6-47.6
1.71 1.67 1.1
η0 (at 200 C), Pa s 1485 2510
of IIqij for the cocontinuous blend are indeed very close to those of the gyroid surface (analytically computed), which suggest that the gyroid can be used as a model surface for cocontinuous morphologies, in terms of anisotropy measurements. The relaxation of the interface after cessation of uniaxial extension is accompanied by a decrease in the anisotropy, as in evidenced in Figure 13. This decrease is manifest as an exponential decay of the second invariant II q ð13Þ 2 ¼ R þ βe - Rt Q with R = 0.017, β = 0.564, and R = 0.129. Similar to the coarsening process in immiscible blends,42 anisotropy relaxation is driven by interfacial energy stored at the curved interface and hindered by viscous forces. Hence, the rate R in eq 13 is proportional to Γ/η0. Further experimental work is required to verify this statement. The offset, R, denotes the anisotropy at t f ¥, i.e., a residual anisotropy that cannot be further relaxed when the viscous forces overshadow the interfacial tension forces. The large error bars in Figure 13 could be result of the big amount of small droplets generated during the time of deformation and retraction, which are not counted in the analysis because their sizes are below the resolution of the 3D rendering algorithm (see Figure 14). Note that the blend used in this study is compatibilized with PSb-PMMA block copolymer that has been proved to hinder the coarsening of the cocontinuous domains.25 This implies that the interfacial tension of this system is considerably lower than that of the noncompatibilized blend, ergo, the interface relaxation of the (42) Lopez-Barron, C. R.; Macosko, C. W. Soft Matter 2010, 6, 2637–2647.
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Figure 11. 3D rendered images of FLPS/SAN blend subjected to uniaxial extension with (a) ε = 0.0, (b) ε = 0.4 and (c) ε = 2.0.
Figure 12. Normalized invariants of the interface tensor as a function of the Hencky strain for the FLPS/SAN blend during uniaxial deformation. Error bars indicate the standard deviation calculated from measurement of three different samples. Solid lines are results for the gyroid surface computed analytically (also shown in Figure 7).
Figure 13. Normalized invariants of the interface tensor as a function of annealing time for the FLPS/SAN blend during relaxation (after uniaxial deformation). The point at annealing time = 0.0 min corresponds to the point at ε = 2.0 in Figure 12. Error bars indicate the standard deviation calculated from measurements of three different samples. The solid line denotes the fitting with the exponential equation |IIq| =Q2(0.017 þ 0.564e-0.129t).
latter is faster. Accordingly, the LCPM proposed here can be used to quantify the effect of interfacial modifiers (such as block copolymers) on the interfacial tension by measuring the evolution of the interface anisotropy of systems with different amounts of compatibilizer. Finally, the retraction of systems with axisymmetric interfaces (e.g., elongated droplets) have been extensively investigated and well-known relations between interfacial tension and the shape43-45 or anisotropy19 of the droplets have been developed. (43) Stone, H. A.; Bentley, B. J.; Leal, L. G. J. Fluid Mech. 1986, 173, 131–158. (44) Elmendrop, J. J. Polym. Eng. Sci. 1986, 26, 418–426. (45) Elemans, P. H. M.; Janssen, J. M. H.; Meijer, H. E. H. J. Rheol. 1990, 34, 1311–1325.
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Figure 14. Upper image: 3D rendered micrograph of FLPS/SAN blend quiescently annealed for 10 min after uniaxial strain (ε = 2.0). Lower image: Detail of a row confocal image showing some small droplets (indicated by the red arrows) that were not reconstructed by the rendering algorithm.
Likewise, the rheological behavior in relation with the morphology during the process of relaxation after the application of large strains has been investigated for blends with droplet-matrix morphology.18,46,47 Similar studies on systems with more complex interfaces (e.g., bicontinuous) will be possible using the LCPM proposed here. (46) Yamane, H.; Takahashi, M.; Hayashi, R.; Okamoto, K.; Kashihara, H.; Masuda, T. J. Rheol. 1998, 42, 567–580. (47) Iza, M.; Bousmina, M. J. Rheol. 2000, 44, 1363–1384.
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Figure 15. Components of the interface tensor qij as a function of (a) shear strain and (b) Hencky strain. The solid lines denote the values computed with (a) eqs 23 and 24 and (b) eq 26, and the symbols give the values calculated by applying the LCPM on the gyroid surface.
Conclusions A method (the local cross product method, LCPM) to quantify the interface anisotropy of two phase systems was developed on the basis of differential geometry coupled with 3D imaging. The method is applicable to systems with complex interfaces provided that a 3D triangular mesh representation of the interface is available. The accurate quantification of the anisotropy is crucial in the computation of interfacial stresses and in the development of constitutive equations, as demonstrated by Doi and Ohta,16 hence the importance of the LCPM. The LCPM was tested with a model bicontinuous structure (the gyroid surface) subjected to uniaxial and shear deformations. Excellent qualitative agreement was found between the components of the interface tensor and its invariants computed analytically and with our method. Quantitative errors within 3% and 5% were found for small values of Hencky strain and shear strain, respectively. More substantial errors were observed for bigger deformations. Furthermore, the anisotropy evolution during uniaxial extension and subsequent relaxation of an immiscible polymer blend with cocontinuous morphology was computed with the LCPM. The evolution of the invariants of qij follows the same trends for both the gyroid surface and the cocontinuous interface. This finding is promising in the sense that the analytically generated gyroid could be used to model anisotropy evolutions of cocontinuous blends subjected to any flow field. Finally, it is noteworthy that the ability to characterize the anisotropy of complex interfaces during retraction could lead to a new method to estimate interfacial energies, due to the direct relation between rate of retraction and interfacial tension. Acknowledgment. This work was supported primarily by the MRSEC Program of the National Science Foundation under Award Number DMR-0212302. Part of this work was carried out in the Institute of Technology Polymer Characterization Facility, University of Minnesota, which has received capital equipment funding from the NSF through the MRSEC. Image acquisition were carried out in the Biomedical Image Processing Laboratory and the image processing and analysis at the Minnesota Supercomputing Institute at the University of Minnesota.
Appendix A Here we derive theoretical expressions for Q/Q0 as a function of λ (uniaxial extension) and γ (shear) using the second invariant of the Finger deformation tensor. 14292 DOI: 10.1021/la102314r
Uniaxial Extension. Considering an incompressible fluid, the deformation gradient tensor for the case of uniaxial deformation is 2 3 λ 0 0 6 7 - 1=2 ð14Þ F ¼ 6 0 7 40 λ 5 - 1=2 0 0 λ and the Finger tensor (B = F 3 F) 2 λ2 0 6 -1 B ¼ 6 40 λ 0 0
0
3
7 0 7 5 λ
ð15Þ
-1
Hence, the second invariant of B (IIB = 1/2[(tr B)2 - tr B2]) is given as 1 ð16Þ IIB ¼ 2λ þ 2 λ It is well known that the square root of the second invariant of B gives the average area change on all the planes around any point embedded in the material subjected to deformation.35 Hence, assuming affine deformation, the interfacial area change due to uniaxial extension can be computed with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi Q IIB 1 1 2λ þ 2 ¼ ð17Þ ¼ Q0 3 3 λ Shear. Similarly, for simple shear, the deformation gradient tensor, the Finger tensor and its second invariant are given as 2 3 1 γ 0 ð18Þ F ¼ 40 1 05 0 0 1 2
3 1 þ γ2 γ 0 B ¼ 4 γ 1 05 0 0 1
ð19Þ
IIB ¼ 3 þ γ2
ð20Þ
and
Thus, Q ¼ Q0
rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi IIB γ2 ¼ 1þ 3 3
ð21Þ
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Equations 17 and 21 were plotted and compared with interfacial area data computed with the LCPM for a gyroid surface (Figure 3). It is remarkable that results from both calculations were in perfect accordance, in spite of the fact that the analyses leading to eqs 17 and 21 do not assume any particular morphology.
Appendix B From the kinetic equations developed by Doi and Ohta to describe the time dependence of qij and Q,16 they proposed expressions for Q/Q0 and components of qij/Q0 as a function of strain for simple shear: Q ¼ Q0
γ2 1þ 3
!1=2
qxz γ γ2 ¼ - 1þ 3 Q0 3
ð22Þ ! - 1=2
qxx - qzz γ2 γ2 1þ ¼ Q0 3 3
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ð23Þ ! - 1=2 ð24Þ
and uniaxial extension: Q ð2eε=2 þ e - 5ε=2 Þ ¼ Q0 ½3ð2 þ e - 3ε Þ1=2
ð25Þ
qzz - qxx ðeε=2 - e - 5ε=2 Þ ¼ Q0 ½3ð2 þ e - 3ε Þ1=2
ð26Þ
The derivation of these expressions can be found in ref 16. Here we compare the quantities computed with eqs 22 to 26 with the corresponding calculated with the LCPM for the gyroid surface (eqs 8, 9 and 10). Notice that the expressions for Q/Q0 for shear from Doi-Ohta theory (eq 22) and from the second invariant (eq 21) are identical. Likewise both equations for Q/Q0 for uniaxial extension (eqs 17 and 25) give the same asymptotic form for ε .1, i.e., Q=Q0 (2/3)1/2eε/2. In fact plots of Q/Q0 vs ε from eqs 17 and 25 overlap on each other (not shown) in the whole strain range studied here. The quantities qxz/Q0 and (qxx - qzz)/Q0 for shear deformation as well as for uniaxial extension computed with the LCPM and from eqs 23, 24 and 26, respectively, are shown in Figure 15. The values measured with the LCPM are notably well predicted by the Doi-Ohta’s expressions for strains γ or ε>1. This result support the validity of the method to compute the interface tensor proposed in this work (the LCPM). The error observed in the very small strain regime could be due to both the resolution limit of the LCPM method and the resolution limit of the 3D images analyzed.
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